We are given the binary operation $*$ defined on the set $\mathbb{R}$ of real numbers by $m * n = m + n + 2$. We need to find both the identity element and the inverse of a number $n$ under this operation.

1. Finding the Identity Element

To find the identity element $e \in \mathbb{R}$, we require:

$m * e = m \quad \text{for all} \quad m \in \mathbb{R}$

Substituting the operation into the equation:

$m * e = m + e + 2 = m$

To solve for $e$, subtract $m$ from both sides:

$e + 2 = 0$

Thus, the identity element is:

$e = -2$

2. Finding the Inverse of $n$

The inverse of a number $n$ under this operation is a number $n' \in \mathbb{R}$ such that:

$n * n' = e$

Given that the identity element $e = -2$, we substitute into the operation:

$n * n' = n + n' + 2 = -2$

To find $n'$, solve the equation:

$n + n' + 2 = -2$

Subtract 2 from both sides:

$n + n' = -4$

Now, solve for $n'$:

$n' = -4 - n$

So, the inverse of $n$ under this operation is:

$n' = -n - 4$

Summary:

• Identity Element: $-2$
• Inverse of $n$: $-n - 4$

Would you like more details or have any questions?

Here are 8 related questions that expand on this topic:

1. How do you verify that $n' = -n - 4$ is indeed the inverse of $n$?
2. Can you find the inverse of $n$ if the operation was $m * n = m + n - 2$?
3. How does the identity element affect the structure of the operation in a group?
4. Is this operation associative? How would you prove or disprove it?
5. What would the inverse of $n = 3$ be under this operation?
6. Can you express the operation $m * n = m + n + 2$ in a different form?
7. What is the result of $5 * (-3)$ under this operation?
8. How do properties of this operation compare to standard addition on $\mathbb{R}$?

Tip: To understand binary operations deeply, always analyze the existence and uniqueness of identity and inverse elements, as these are key properties in the context of group theory.